Adaptive system in a digital data receiver providing compensation for amplitude and phase distortions introduced by a data transmission channel

ABSTRACT

An adaptive system in a digital data receiver providing compensation for amplitude and phase distortions introduced by the data transmission channel having, at the output of the transmission channel which supplies a signal vector X k , a transversal filter having N weighting coefficients, followed by a decision circuit and further a summing circuit and a multiplying circuit. To determine the N coefficients iteratively the system comprises a circuit for determining an estimated matrix A k  of the autocorrelation matrix A of the signal vectors X k , a circuit for approximating A k  by a circulant matrix R k , a circuit for calculating the diagonal matrix G k  whose diagonal elements are the eigenvalues of R k , a number of calculating circuits and a circuit for up-dating the vector C k  which represents the N weighting coefficients of the filter at the instant t o  +kT.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an adaptive system in a digital datareceiver providing compensation for amplitude and phase distortionsintroduced by a data transmission channel.

2. Description of the Prior Art

It is a known fact that at high transmission rates a correctrestitution, at the output of the data transmission channel, of thesignals applied to its input is no longer possible without the provisionof a compensation circuit, designated equalizer, which is oftenconstituted by a non-recursive transversal filter, that is to say acircuit capable of correcting the response of a transmission channel onthe basis of a finite and weighted sum of partial responses available onthe consecutive taps of an impedance network based on delay lines. Anequalizer of a conventional type, having N weighting coefficients isshown in FIG. 1 (a description of such an equalizer having sevencoefficients is given in the publication IEEE Transactions onInformation Theory, Vol. IT-15, No. 4, July 1969, pages 484 to 497).Since the impulse response of the channel is not known and furthermoretends to evolve in time, the equalizer must be adaptive, that is to sayit must be capable of adjusting its weighting coefficients to theiroptimum values at the beginning of the transmission (during theacquisition or training phase of the equalizer) and of followingthereafter any variations of the channel during the actual transmissionphase. This adaptivity finds expression in that the equalizer generatesan error signal which is a function of the difference between the exactform of the transmitted digital data and the form they have to theoutput of the equalizer, and is arranged so as to reduce this error to aminimum.

In order to provide an efficient use of the receiving system, thetraining phase must be as short as possible, which means that the methodof determining the optimum coefficients of the adaptive equalizer mustconverge as rapidly as possible. Because of their aptitude in followingthe temporal variations of the data transmission channel, iterativedetermination methods are frequently employed, such as the stochasticmethod. But the convergence speed of this method decreases according asthe eigenvalues of the signal autocorrelation matrix A of the outputsignal of the transmission channel are more dispersed, that is to sayaccording as the amplitude distortion introduced by the channel is moreimportant. If the channel were perfect and its spectrum perfectly flat,the distortion would be zero and all the eigenvalues of A would be equalto 1. In reality, as soon as the channel introduces a significantamplitude distortion (or as soon as intersymbol interference isdeliberately created for spectrum shaping purposes), the use of thestochastic gradient method becomes ineffectual.

A satisfactory convergence rate may be obtained by using an iterativeequalization method of the self-orthogonalizing type as described in thearticle by R. D. Gitlin and F. R. Magee "Self-Orthogonalizing AdaptiveEqualization Algorithms", published in IEEE Transactions onCommunications, Vol. COM-25, No. 7, July 1977, pages 666 to 672, theKalman filter (applied to the field of equalization by D. Godard, see inthis respect reference [16] on page 672 of said article) constituting aspecial case of this method. However, compared to the stochasticgradient method this novel method is characterized by the fact that thecircuits required for its implementation are much more complicated andby the fact that the number of operations to be performed in the courseof each iteration is increased considerably.

SUMMARY OF THE INVENTION

The invention has for its object to provide an adaptive system in adigital data receiver, in which system the determination of the valuesof the coefficients of the equalizer by consecutive iterations isrealized at a convergence speed which is almost as high as in thelast-mentioned article, but whose circuits are however much simpler.

In accordance with the present invention, there is provided, in adigital data receiver, an adaptive system providing compensation foramplitude and phase distortions introduced by a data transmissionchannel and comprising an adaptive equalizing circuit receiving a signalvector X_(k) from the data transmission channel output and producing anoutput signal y_(k), a decision circuit receiving this output signaly_(k) and producing an estimation s_(k-d) of each one of the digitaldata s_(k-d) applied to the data transmission channel input, a summingcircuit receiving said output signal y_(k) and said estimation s_(k-d)for producing a difference signal e_(k) =y_(k) -s_(k-d), and amultiplying circuit for multiplying said difference signal e_(k) by aniteration step α_(k), characterized in that the adaptive equalizingcircuit is a nonrecursive transversal filter having N adjustableweighting coefficients and the adaptive systems comprises for thedetermination of these coefficients by consecutive iterations:

a first circuit for determining, in a matrix A_(k) which is anestimation of the square signal autocorrelation matrix A=E(X_(k) ·X_(k)^(TR)) at an instant t_(o) +kT, where E is the expectation operator,X_(k) ^(TR) is the transpose of X_(k), t_(o) is a constant, k is aninteger and T is the duration of a data symbol period, the (N/2+1) firstelements a_(i).sup.(k) of the first row if N is even, or the (N+1)/2first elements a_(i).sup.(k) of the first row if N is odd, saidestimation being based on the relation:

    a.sub.i.sup.(k) =βa.sub.i.sup.(k-1) +x.sub.k.sup.TR ·x.sub.k-i

where i is an integer with 0≦i≦N-1 and β is a constant with 0<β<1,

a second circuit connected to said first circuit for forming a vectorU.sup.(k) =[r₀.sup.(k), r₁.sup.(k), r₂.sup.(k), . . . , r_(N-2).sup.(k),r_(N-1).sup.(k) ]^(TR) in which r_(i).sup.(k) =a_(i).sup.(k) for every inot exceeding N/2 if N is even, or not exceeding (N-1)/2 if N is odd,and r_(i).sup.(k) =a_(N-i).sup.(k) for every i exceeding said limits,

a third circuit connected to said second circuit for forming a vector

    Λ.sup.(k) =[λ.sub.0.sup.(k), λ.sub.1.sup.(k), λ.sub.2.sup.(k), . . . , λ.sub.N-1.sup.(k) ].sup.TR,

whose components are the eigenvalues of a circulant matrix having saidvector U.sup.(k)TR as a first row, with the aid of the relation:##EQU1## where P is the unitary matrix of the order N defined by:

    P=∥P.sub.f,g ∥

(f,g=0, 1, 2, . . . , N-2, N-1)

and ##EQU2## a fourth circuit connected to the data transmission channeloutput and the multiplying circuit output for forming a vector Q.sup.(k)with the aid of the relation:

    Q.sup.(k) =α.sub.k e.sub.k ·P.sup.cc ·X.sub.k

where P^(cc) is the complex conjugate of said unitary matrix P,

a fifth circuit connected to said fourth and third circuits for dividingsaid vector Q.sup.(k) by said vector Λ.sup.(k) on a term-by-term basisso that the resulting vector F.sup.(k) =[f₀.sup.(k), f₁.sup.(k),f₂.sup.(k), . . . , f_(N-1).sup.(k) ]^(TR) =Q.sup.(k) /Λ.sup.(k) hascomponents f_(i).sup.(k) =q_(i).sup.(k) /λ_(i).sup.(k) for every i,

a sixth circuit connected to said fifth circuit for multiplying saidvector F.sup.(k) by said unitary matrix for producing a vector H.sup.(k)=P·F.sup.(k),

a seventh circuit connected to said sixth circuit and said transversalfilter for up-dating the vector C_(k) of the N weighting coefficients ofsaid transversal filter at the instant t_(o) +kT so as to produce acoefficient vector C_(k-1) at the instant t_(o) +(k+1)T in accordancewith the relation:

    C.sub.k+1 =C.sub.k -H.sup.(k)

In a second embodiment of the adaptive system according to the presentinvention, the system further comprises:

an eighth circuit connected to the data transmission channel output formultiplying said signal vector X_(k) by said unitary matrix P, theresulting signal vector Z_(k) =P·X_(k) being applied to said transversalfilter input,

a ninth circuit interconnecting said sixth and seventh circuits formultiplying said vector H.sup.(k) at the output of said sixth circuit bysaid matrix P^(cc) and applying the resulting vector P^(cc) ·H.sup.(k)to said seventh circuit, said seventh circuit being arranged forup-dating the vector D_(k) =P^(cc) ·C_(k) representing the N weightingcoefficients of said transversal filter at the instant t_(o) +kT whenthis filter receives said signal vector Z_(k), so as to produce acoefficient vector D_(k+1) =P^(cc) ·C_(k+1) at the instant t_(o) +(k+1)Tin accordance with the relation:

    D.sub.k+1 =D.sub.k -P.sup.cc ·H.sup.(k)

BRIEF DESCRIPTION OF THE DRAWINGS

Further particulars of the invention will be apparent from the followingdetailed description with reference to the accompanying drawings inwhich:

FIG. 1 shows a non-recursive transversal filter of a known type, havingN weighting coefficients;

FIG. 2a shows the signal autocorrelation matrix A of N consecutivesamples of the random process x(t) to which the consecutive outputsignal vectors X_(k) of the data transmission channel correspond, FIG.2b shows a simplified form of this matrix A taking account of the factthat x(t) is a random process of the stationary type, and FIG. 2c showsa still further simplified form of the matrix A taking account of thelength of the sampled response of the channel;

FIG. 3a represents, for the case where N is odd and equal to 2S+1, thecirculant matrix R chosen, on the basis of FIG. 2b, so as to constitutethe approximation of the matrix A, and FIG. 3b shows a simplified formof said matrix R taking account of the length l of the sampled responseof the channel;

FIG. 4 shows which partition may be effected in the structure of thematrix product of the inverse of the matrix R by the matrix A;

FIG. 5 shows a first embodiment of the adaptive system in accordancewith the invention;

FIG. 6a and FIG. 6b show two variants of the structure of a secondembodiment of the adaptive system in accordance with the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

To describe the two embodiments of the system in accordance with theinvention, specified more in detail hereinafter, it is first assumedthat the transmitter system provided at the beginning of thetransmission channel transmits a stream of non-correlated binary datas_(k), equal to ±1, that the impulse response of the transmissionchannel, is h (t), and that the receiver is in perfect synchronizationwith the transmitter.

If x(t) is the signal received and sampled at a rate of 1/T (T being theduration of a data symbol) and if the sample entering the equalizer atan instant t_(o) +kT is defined by the relation: ##EQU3## where l is thelength of the sampled response of the transmission channel and n_(k) thenoise at the instant t_(o) +kT, it is possible to define for an adaptiveequalizer having N weighting coefficients c₀, c₁, c₂, . . . c_(N-2),c_(N-1) (see FIG. 1) the following column vectors, which for the sake ofconvenience are written in the equivalent form of the transpose rowvectors (TR indicating the transposition operator):

    X.sub.k =[x.sub.k, x.sub.k-1, . . . , x.sub.k-N+2, x.sub.k-N+1 ].sup.TR(2)

    C=[c.sub.0, c.sub.1, c.sub.2, . . . , c.sub.N-2, c.sub.N-1 ].sup.TR(3)

By definition, the output signal of the equalizer, before decision, is:

    y.sub.k =C.sup.TR ·X.sub.k                        (4)

or, in an equivalent form:

    y.sub.k =X.sub.k.sup.TR ·C                        (4 bis)

and the difference, at the instant defined by k, between y_(k) and thecorresponding digital data s_(k-d) is given by the relation:

    e.sub.k =C.sup.TR ·X.sub.k -s.sub.k-d

or

    e.sub.k =X.sub.k.sup.TR ·C-s.sub.k-d              (5)

It should be noted that the decision about each transmitted symbol isproduced with a delay d·T with respect to the transmission of saidsymbol, which delay d·T can be optimized for minimizing the mean squareerror, in the manner indicated more in particular in an article by P.Butler and A. Cantoni, "Non-iterative Automatic Equalization", publishedin IEEE Transactions on Communications, Vol. COM-23, No. 6, page 622, §III-A, lines 5-6, and page 624, § C, of said article. It is assumed thatall decisions are correct, i.e. S_(k-d) =S_(k-d) for every value of k.

In the equalizer art there are different criteria to reduce thisdifference between the exact form of the transmitted signals and theestimated form they have at the output of the equalizer. Here we shallconsider one of the most frequently used criteria, namely the criterionof minimizing the mean square error, and we shall therefore try tochoose the vector C in order to minimize the mean value of e_(k) ².

Using the two equivalent relations (4) and (4 bis) simultaneouslyresults in that:

    e.sub.k.sup.2 =(C.sup.TR ·X.sub.k -s.sub.k-d) (X.sub.k.sup.TR ·C-s.sub.k-d)

    e.sub.k.sup.2 =C.sup.TR ·X.sub.k ·X.sub.k.sup.TR ·C-2C.sup.TR ·X.sub.k s.sub.k-d +1      (6)

The mean value of e_(k) ² is taken, namely:

    E(e.sub.k.sup.2)=C.sup.TR ·A·C-2C.sup.TR ·V+1(7)

where E is the expectation operator and:

    A=E(X.sub.k ·X.sub.k.sup.TR)                      (8)

    V=E(X.sub.k ·s.sub.k-d)                           (9)

The relation (7) defines the mean square error for a given vector C,which error should be minimized as a function of C. For this purpose, itis necessary that: ##EQU4## G(C) is zero if AC=V, that is to say if:

    C=A.sup.-1 V                                               (10)

The vector C searched for might therefore be obtained by directresolution of the equation (10). This resolution is considered to bedifficult, for it implies the inversion of the matrix A, that is to saythe realization of a very large number of operations, but a novel methodof direct determination of an approximated vector of C, which avoidssuch a complex operation is proposed and described in Applicants'copending U.S. patent application Ser. No. 260,617, filed on May 15,1981 too.

However, in the foregoing it was mentioned for what reasons iterativemethods are used more often. According to the invention which is thesubject of the present application and which relates to a noveliterative method of determination of the vector of the weightingcoefficients of the equalizer, the expression of the following iterativealgorithm will first be considered:

    C.sub.k+1 =C.sub.k -α.sub.k ·A.sub.k.sup.-1 ·X.sub.k e.sub.k                                 (11)

which algorithm is of the type employed in the transversal equalizer ofthe receiving system described in the article by Gitlin and Mageementioned in the foregoing, and in which:

C_(k) =vector C at the instant t_(o) +kT

C_(k+1) =vector C at the instant t_(o) +(k+1)T

α_(k) =iteration step (fixed or variable)

A_(k) ⁻¹ =the inverse of the matrix A_(k) defined by: ##EQU5## A_(k) isthe estimated matrix, at the instant t_(o) +kT, of the signalautocorrelation matrix A.

In the expression (11), the estimating A_(K) ⁻¹ must be calculated ateach iteration in accordance with the following relation: ##EQU6## Therelation (12) clearly shows the complexity of the solution adopted inthe system described in the article by Gitlin and Magee. However, asA_(k) ⁻¹ rapidly converges to A⁻¹, it is possible to avoid thiscomplexity by searching for an approximation of the matrix A⁻¹, andconsequently of the matrix A. Such an algorithm is based on thefollowing observations: the matrix A defined by the relation (8) andshown in FIG. 2a is the autocorrelation matrix of N consecutive samplesof the random process x(t). As this process is of the stationary type,all the elements E(X_(i) ²) are equal; for the same reason, all theelements E(X_(i) ·X_(j)) for which |i-j| is constant are also equal. Asa result thereof, matrix A is symmetrical, its diagonal elements areequal and it can therefore be written in the form shown in FIG. 2b. Onthe other hand as l is the length of the impulse response of thechannel, all the elements E(X_(i) ·X_(j)) in which the differencebetween i and j is equal to or higher than l are zero, as theycorrespond to received signals between which there is no correlationwhatsoever. Ultimately, the matrix A therefore takes the simplified formshown in FIG. 2c.

This matrix A is quasi-diagonal (and also its inverse matrix A⁻¹, butfor marginal effects) and it is possible to define an approximation,which is better according as the number N of the equalizer coefficientsis chosen greater with respect to the length l. This approximation is acirculant matrix R which is constructed in the following way. If (a₀,a₁, a₂, . . . a_(N-2), a_(N-1)) is the first row of A and (r₀, r₁, r₂ .. . r_(N-1)) the first row of R, the element r_(i) is defined by r_(i)=a_(i) for every i which is lower than or equal to N/2 if N is even, orto (N-1)/2 if N is odd, and by r_(i) =a_(N-i) for every i which ishigher than these limits.

The following rows of R, of the order 2 to N, comprise the same elementsas the first row of R, but after cyclic permutation of these elements tothe right: after a cyclic permutation to the right for the second rowwith respect to the first row, after a further cyclic permutation to theright for the third row with respect to the second row, and so on up tothe N^(th) row of R. For the case that N is odd and equal to (2S+1), thematrix R thus obtained is shown in FIG. 3a. Just like the matrix A, saidmatrix R may assume the simplified form shown in FIG. 3b andcorresponding to the simplified form of A shown in FIG. 2c.

As A_(k) ⁻¹ rapidly converges to A⁻¹, the expression (11) becomes:

    C.sub.k+1 =C.sub.k -α.sub.k A.sup.-1 X.sub.k e.sub.k (12)

and, taking the average value, relation (12) becomes:

    E(C.sub.k+1)=E(C.sub.k)-α.sub.k A.sup.-1 E(X.sub.k e.sub.k)(13)

In accordance with the relations (4), (4 bis) and (5):

    e.sub.k =X.sub.k.sup.TR ·C.sub.k -s.sub.k-d

hence

    E(X.sub.k e.sub.k)=E(X.sub.k X.sub.k.sup.TR)C.sub.k -E(X.sub.k s.sub.k-d)

    E(X.sub.k e.sub.k)=AC.sub.k -V

    E(X.sub.k e.sub.k)=AC.sub.k -AC=A(C.sub.k -C)=AC.sub.k     (14)

where we have used the notation C_(k) =C_(k) -C and by transferring thisresult to relation (13), the latter becomes:

    E(C.sub.k+1)=E(C.sub.k)-α.sub.k C.sub.k              (15)

This relation (15) furnishes at each iteration the optimum adjustmentdirection and, consequently, converges very rapidly. The importance ofreplacing A⁻¹ by a matrix which constitutes a proper approximation, thusby R⁻¹ in the present case, in order to avoid a complicated procedure toobtain the estimated matrix of A⁻¹, will be evident. The calculation ofR⁻¹. A undertaken to justify this approximation actually shows that thematrix product has the shape shown in FIG. 4 and that it comprises:

a kernel which is identical to the identity matrix of the orderN-2(l-1);

zero elements above and below said identity matrix; and

on either side of the columns which include said identity matrix andsaid zero elements, 2(l-1) columns which contain non-zero elements.

Taking account of this structure of the product R⁻¹ ·A and of thepartition effected here in order to show its particular character (andtaking account of the fact that the mathematical study of the matrix Rshows that this matrix R is always defined and that its inverse matrixalways exists), the matrix R as defined in the foregoing isasymptotically equivalent to the matrix A and constitutes a properapproximation (satisfactory even if the number N of the coefficients isnot very large).

After having justified in this way the approximation of A by R, the factthat A is the limit of its estimate A_(k) at the instant t_(o) +kTresults in that R is also the limit of its estimate R_(k) at the instantt_(o) +kT. The expression (12), may be replaced by:

    C.sub.k+1 =C.sub.k -α.sub.k R.sup.-1 X.sub.k e.sub.k (16)

and expression (11) by:

    C.sub.k+1 =C.sub.k -α.sub.k R.sub.k.sup.-1 X.sub.k e.sub.k(17)

Diagonalizing the matrix R_(k) makes it possible to write:

    R.sub.k.sup.-1 =PG.sub.k.sup.-1 P.sup.cc                   (18)

where:

G⁻¹ =diagonal matrix whose diagonal elements are the inverse values ofthe eigenvalues λ₀.sup.(k), λ₁.sup.(k), λ₂.sup.(k), . . . , ##EQU7## ofthe matrix R_(k) ; P=the simmetrical unitary matrix of the order N,which is known a priori as this matrix is independent of R_(k) andcommon to all the circulant matrices and whose columns are theeigenvectors of the matrix R_(k) (this matrix P may be defined by:##EQU8## and the multiplication of a vector by this matrix produces, butfor the coefficient 1/√N the inverse Discrete Fourier Transform, or theinverse DFT, of this vector); and where:

P^(cc) =the complex conjugate matrix of P (in a similar manner, saidmatrix P^(cc) may be defined by: ##EQU9## and multiplying a vector byP^(cc) produces, but for the coefficient 1/√N, the Discrete FourierTransform, or DFT, of this vector).

By transferring the expression (18) to the expression (17) it becomes:

    C.sub.k+1 =C.sub.k -α.sub.k PG.sub.k.sup.-1 P.sup.cc X.sub.k e.sub.k( 19)

It will be seen that from now onwards the vector ##EQU10## of the Neigenvalues of R_(k) is obtained by means of the following relation:##EQU11## in which P has already been defined and U.sup.(k) is thecolumn vector of the N elements of the first column of R_(k).

From the expression (19) it is possible to derive a first structure ofthe digital data receiving system, which structure permits of obtainingthe vector of the optimum weighting coefficients for the adaptive filterof this system by consecutive iterations. This system is shown in FIG. 5and comprises at the output of a data transmission channel 1 anon-recursive adaptive filter 2, designated F.A., which has adjustableweighting coefficients, and receives the output signal vector X_(k) ofthe transmission channel. This filter is followed by a decision circuit3, designated C.D., which produces from the output signal y_(k) of thefilter an estimation s_(k-d) of each of the digital data s_(k-d) appliedto the input of the channel (the system knows the transmitted digitaldata only during the acquisition or training phase and only then theestimation s_(k-d) is always equal to the data s_(k-d) itself). Asumming circuit 4a, which receives y_(k) and s_(k-d), produces adifference signal e_(k) =y_(k) -s_(k-d) which represents the differencebetween the output of the filter and the estimation of the transmitteddata; a multiplying circuit 4b which multiplies e_(k) by the step α_(k)produces the scalar quantity α_(k) e_(k).

For the determination of the weighting coefficients of the filter 2 thesystem shown in FIG. 5 also comprises:

(A) an estimation circuit 5 for the square matrix A_(k), which matrix isan estimate of the autocorrelation matrix of the signal vectors X_(k) atthe instant t_(o) +kT. It should be noted that the estimation of A_(k)does not signify the estimation of N² elements, but simply, in its firstrow, the estimation of the (N/2+1) first elements a_(i).sup.(k) if N iseven, or of the (N+1)/2 first elements a_(i).sup.(k) if N is odd, on thebasis of the relation

    a.sub.i.sup.(k) =βa.sub.i.sup.(k-1) +x.sub.k ·x.sub.k-i

where β is a constant located between the values 0 and 1 but notincluding these values. This important reduction of the number of theelements to be estimated is very advantageous as regards the simplicityof the circuits.

(B) a circuit 6 for approximating the matrix A_(k) by a circulant matrixR_(k), this approximation of A_(k) by R_(k) being obtained in a simpleway by substituting the first row ##EQU12## in which r_(i) =a_(i) forevery i not exceeding N/2 if N is even, or not exceeding (N-1)/2 if N isodd, and r_(i) =a_(N-i) for every i higher than these limits. As R_(k)is circulant, the determination of its first row is sufficient to knowthe entire matrix, and the approximation of A_(k) by R_(k) amountstherefore to not more than the formation of the vector ##EQU13## forevery i not exceeding N/2 if N is even, or not exceeding (N-1)/2 if N isodd, and ##EQU14## for every i higher than these limits.

(C) a circuit for calculating the diagonal matrix G_(k), whose diagonalterms are the eigenvalues of the circulant matrix R_(k), which circuit,taking account of the simplifications already effected sub (A) and (B),actually consists of a circuit 7 for forming a vector Λ.sup.(k) of Neigenvalues λ₀.sup.(k), λ₁.sup.(k), λ₂.sup.(k), . . . ##EQU15## by meansof the relation Λ.sup.(k) =√N·P·U.sup.(k), in which P is the unitarymatrix which has already been defined.

(D) a circuit 8 for forming the vector Q.sup.(k) =α_(k) e_(k) P^(cc)X_(k) which circuit 8 ensures in fact the multiplication of the signalvector X_(k) by the complex conjugate matrix P^(cc) of P and themultiplication of the vector thus obtained by the scalar quantity α_(k)e_(k).

(E) a circuit for multiplying the output of circuit 8 by the diagonalmatrix G_(k) ⁻¹, which circuit actually consists, because of thesimplifications already effected, of a circuit 9 for dividing the vectorQ.sup.(k) by the vector Λ.sup.(k) on a term-by-term basis so that theresulting vector ##EQU16## has components of f_(i).sup.(k)=q_(i).sup.(k) /λ_(i).sup.(k), whatever the value of i.

(F) a circuit 10 for multiplying the vector F.sup.(k) thus obtained bythe matrix P.

(G) a circuit 11 for calculating the vector C_(k+1) =C_(k) -P·F.sup.(k)=C_(k) -H.sup.(k) which, at the instant t_(o) +(k+1)T, represents thevector of the N weighting coefficients of the transversal filter 2 andwhich is obtained by the difference between the vector C_(k), evaluatedin a similar way in the course of the preceding iteration stepcorresponding to the instant t_(o) +kT, and the vector H.sup.(k) at theoutput of the circuit 10; this circuit 11 is arranged in a conventionalway for up-dating the coefficients of filter 2 by substituting C_(k+1)for C_(k).

A second structure of the receiving system in accordance with theinvention can be derived from the expression (19) obtained above, if amultiplication of the two members of this expression by P^(cc) l iseffected. Then there is obtained:

    P.sup.cc C.sub.k+1 =P.sup.cc C.sub.k -α.sub.k P.sup.cc PG.sub.k.sup.-1 P.sup.cc X.sub.k e.sub.k                  (21)

    D.sub.k+1 =D.sub.k -α.sub.k G.sub.k.sup.-1 P.sup.cc X.sub.k e.sub.k(21 bis)

In FIG. 6a this multiplication of the expression (19) by P^(cc) results,with respect to FIG. 5, in the appearance of the two supplementarycircuits 12 and 13. The circuit 12 ensures the multiplication of theoutput signal vector X_(k) of the transmission channel 1 by the unitarymatrix P in order to supply a singal vector Z_(k) which is applied tothe transversal filter 2, and the circuit 13, which is inserted betweenthe circuits 10 and 11 ensures the multiplication of the output of thecircuit 10 by the matrix P^(cc). As the output signal of the filter, inthe case of the above-described first structure, was given by ##EQU17##(in accordance with the relation 4 bis) and this expression may also bewritten as follows: ##EQU18## it is obtained that: ##EQU19## In orderthat the output signal of the filter 2 in the course of thecorresponding iteration at the instant t_(o) +(k+1)T will be the same asin the case of the first structure, the circuit 11 must therefore nowproduce the vector D_(k+1) =P^(cc) ·C_(k-1) obtained from the differencebetween the vector D_(k), evaluated in the course of the precedingiterative stage corresponding to the instant t_(o) +kT, and the vectorof the output P^(cc) ·H.sup.(k) of the circuit 13.

From FIG. 6a corresponding to the second structure thus described it canbe seen that, with the introduction of the circuit 13, the circuits 10and 13 effect consecutively operations which are the inverse of eachother (multiplication by the matrix P, thereafter multiplication by thematrix P^(cc)). These two circuits may therefore both be omitted, whichresults in the variant of the simplified embodiment of FIG. 6b.

It will be understood that further embodiments of the invention maystill be proposed without departing from the scope of the invention.

What is claimed is:
 1. In a digital data receiver, an adaptive systemproviding compensation for amplitude and phase distortions introduced bya data transmission channel and comprising an adaptive equalizingcircuit receiving a signal vector X_(k) from the data transmissionchannel output and producing an output signal y_(k), a decision circuitreceiving this output signal y_(k) and producing an estimation s_(k-d)of each one of the digital data s_(k-d) applied to the data transmissionchannel input, a summing circuit receiving said output signal y_(k) andsaid estimation s_(k-d) for producing a difference signal e_(k) =y_(k)-s_(k-d), and a multiplying circuit for multiplying said differencesignal e_(k) by an iteration step α_(k), characterized in that theadaptive equalizing circuit is a non-recursive transversal filter havingN adjustable weighting coefficients and the adaptive systems comprisesfor the determination of these coefficients by consecutive iterations:afirst circuit for determining, in a matrix A_(k) which is an estimationof the square signal autocorrelation matrix A=E(X_(k)·X_(k).sbsb.TR.sup. TR) at an instant t_(o) +kT, where E is theexpectation operator, X_(k) is the transpose of X_(K), t_(o) is aconstant, k is an integer and T is the duration of a data symbol period,the (N/2+1) first elements a_(i).sbsb.(k) of the first row if N is even,or the (N+1)/2 first elements a_(i) of the first row if N is odd, saidestimation being based on the relation:

    a.sub.i.sup.(k) =βa.sub.i.sup.(k-1) +x.sub.k ·x.sub.k-i

where i is an integer with 0≦i≦N-1 and β is a constant with 0<β<1, asecond circuit connected to said first circuit for forming a vector##EQU20## in which r_(i).sup.(k) =a_(i).sup.(k) for every i notexceeding N/2 if N is even, or not exceeding (N-1)/2 if N is odd, and##EQU21## for every i exceeding said limits, a third circuit connectedto said second circuit for forming a vector ##EQU22## whose componentsare the eigenvalues of a circulant matrix having the transpose of thesaid vector U.sup.(k) as a first row, with the aid of the relation:##EQU23## where P is the unitary matrix of the order N defined by:

    P=|P.sub.f,g |

(f.g=0, 1, 2, . . . , N-2, N-1)and ##EQU24## a fourth circuit connectedto the data transmission channel output and the multiplying circuitoutput for forming a vector Q.sup.(k) with the aid of the relation:

    Q.sup.(k) =α.sub.k e.sub.k ·P.sup.cc ·X.sub.k

where P^(cc) is the complex conjugate of said unitary matrix P, a fifthcircuit connected to said fourth and third circuits for dividing saidvector Q.sup.(k) by said vector Λ.sup.(k) on a term-by-term basis sothat the resulting vector ##EQU25## has componentsf_(i).sup.(k)=q_(i).sup.(k) /λ_(i).sup.(k) for every i not exceedingN-1, a sixth circuit connected to said fifth circuit for multiplyingsaid vector F.sup.(k) by said unitary matrix for producing a vectorH.sup.(k) =P·F.sup.(k), a seventh circuit connected to said sixthcircuit and said transversal filter for up-dating the vector C_(k) ofthe N weighting coefficients of said transversal filter at the instantt_(o) +kT so as to produce a coefficient vector C_(k+1) at the instantt_(o) +(k+1)T in accordance with the relation:

    C.sub.k+1 =C.sub.k -H.sup.(k)


2. An adaptive system as claimed in claim 1, characterized in that thesystem further comprises:an eighth circuit connected to the datatransmission channel output for multiplying said signal vector X_(k) bysaid unitary matrix P, the resulting signal vector Z_(k) =P·X_(k) beingapplied to said transversal filter input, a ninth circuitinterconnecting said sixth and seventh circuits for multiplying saidvector H.sup.(k) at the output of said sixth circuit by said matrixP^(cc) and applying the resulting vector P^(cc) ·H.sup.(k) to saidseventh circuit, said seventh circuit being arranged for up-dating thevector D_(k) =P^(cc) ·C_(k) representing the N weighting coefficients ofsaid transversal filter at the instant t_(o) +kT when this filterreceives said signal vector Z_(k), so as to produce a coefficient vectorD_(k+1) =P^(cc) ·C_(k+1) at the instant t_(o) +(k+1)T in accordance withthe relation:

    D.sub.k+1 =D.sub.k -P.sup.cc ·H.sup.(k)